Cuts of an ordered abelian group
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918225139204096 |
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| author | Vaquié, Michel |
| author_facet | Vaquié, Michel |
| contents | In this article, we study the cuts of a totally ordered abelian group $Γ$. We begin by recalling some results on ordered sets I and on the associated sets IS(I) and FS(I) of initial and final segments of I. For a totally ordered set I we review the notion of an I-structure defined on a module over a ring R, and the definition of the Hahn product of a family of R-modules indexed by I. The set Cv($Γ$)of convex subgroups of a totally ordered group $Γ$ is also a totally ordered set, canonically isomorphic to the set of cuts of the subset Pr($Γ$)of principal convex subgroups. One of the first results is then to equip the group $Γ$ with an I-structure where I is the set Pr($Γ$) endowed with the opposite order. We associate a convex subgroup with every cut of the group $Γ$, and conversely, we can associate a family of cuts with every convex subgroup of $Γ$. It is by looking at these subgroups, and the I-structure of $Γ$ that we can obtain a classification of the different types of cuts. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02382 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cuts of an ordered abelian group Vaquié, Michel Commutative Algebra In this article, we study the cuts of a totally ordered abelian group $Γ$. We begin by recalling some results on ordered sets I and on the associated sets IS(I) and FS(I) of initial and final segments of I. For a totally ordered set I we review the notion of an I-structure defined on a module over a ring R, and the definition of the Hahn product of a family of R-modules indexed by I. The set Cv($Γ$)of convex subgroups of a totally ordered group $Γ$ is also a totally ordered set, canonically isomorphic to the set of cuts of the subset Pr($Γ$)of principal convex subgroups. One of the first results is then to equip the group $Γ$ with an I-structure where I is the set Pr($Γ$) endowed with the opposite order. We associate a convex subgroup with every cut of the group $Γ$, and conversely, we can associate a family of cuts with every convex subgroup of $Γ$. It is by looking at these subgroups, and the I-structure of $Γ$ that we can obtain a classification of the different types of cuts. |
| title | Cuts of an ordered abelian group |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2511.02382 |